I am reading a book about stochastic modelling and I came across something and I couldn't really figure it out. First question would be are Probabilty Generating Functions (PGF) only for discrete variables? Because I have not read that anywhere but overall it always states that discrete variables have a PGF.
Given in that book is a stochastic variable N that is non negative and has a Probabilty Generating Function $P_N(z)$. There is also a variable $Y:= \sum_{j=0}^{N} X_j$ with all $X_j$ iid. $X_j \sim F(\cdot).$ The author of the book then says that you can simply calculate the LaPlace stieltjes tranform of variable $Y$.
So Laplace stieltjes transform would be: $\hat{F}_Y (s) := \mathbb{E}[e^{(-sY)}]. $ My question would be now how am I supposed to go about this the right way? What I did was:
$\mathbb{E}[\mathbb{E}[e^{-s(x_1 + x_2 + ....+ x_N)}|N=n]]$ = $\sum_{n=0}^\infty \mathbb{E}[e^{-s(x_1 + x_2 + ....+ x_N)}] \mathbb{P}(N=n).$
Im not sure this is correct at all, any hint or advice would be nice.
Edit: After spending more time on it I have come to this:
$\displaystyle \sum_{n=0}^\infty \left( \int_0^\infty e^{-st} dF(t) \right)^n \frac{P^{(n)}(0)}{n!} $
Is this correct?
Let's define $\varphi(s) \equiv \mathbb{E}(e^{-s X_1})$ and $\zeta(s) \equiv \mathbb{E}(s^N)$. The Laplace transform of $Y$ is then,
$$ \begin{align} \mathbb{E}(e^{-s Y}) &= \mathbb{E}(e^{- s \sum_{i=1}^{N} X_i}) \\ &= \mathbb{E}[\mathbb{E}( e^{- s \sum_{i=1}^{N} X_i} \mid N)] \\ &= \mathbb{E} \left [ \prod_{i=1}^{N} \mathbb{E}(e^{- s X_i}) \right ] \\ &= \mathbb{E} \left [ \mathbb{E}(e^{- s X_1})^N \right ] \\ &= \mathbb{E} \left [ \varphi(s)^N \right ] \\ &= \zeta [ \varphi(s) ] . \end{align} $$